^ = use e as base RelatedScientific steustatiushistory.org | Log steustatiushistory.org | Root steustatiushistory.org

### What is an exponent?

Exponentiation is a mathematical procedure, created as an, entailing the base a and also an exponent n. In the case wright here n is a positive integer, exponentiation coincides to recurring multiplication of the base, n times.

You are watching: What is 2 to the power of 6

an = a × a × ... × a n times

The steustatiushistory.org over accepts negative bases, however does not compute imaginary numbers. It likewise does not accept fractions, yet can be used to compute fractional exponents, as long as the exponents are input in their decimal create.

### Basic exponent regulations and also rules

When exponents that share the same base are multiplied, the exponents are added.

an × am = a(n+m)EX:22 × 24 = 4 × 16 = 64 22 × 24 = 2(2 + 4) = 26 = 64

When an exponent is negative, the negative authorize is rerelocated by reciprocating the base and also elevating it to the positive exponent.

 a(-n)= 1 an
 EX: 2(-3) = 1 ÷ 2 ÷ 2 ÷ 2 = 1 8
 EX: 2(-3)= 1 23
=1
8

When exponents that share the same base are divided, the exponents are subtracted.

 am an
= a(m - n)
 EX: 22 24
=4
16
=1
4
 22 24
= 2(2-4) = 2-2 =1
22
=1
4

When exponents are increased to one more exponent, the exponents are multiplied.

(am)n = a(m × n)EX: (22)4 = 44 = 256(22)4 = 2(2 × 4) = 28 = 256

When multiplied bases are elevated to an exponent, the exponent is distributed to both bases.

(a × b)n = an × bnEX: (2 × 4)2 = 82 = 64(2 × 4)2 = 22 × 42 = 4 × 16 = 64

Similarly, when divided bases are increased to an exponent, the exponent is dispersed to both bases.

 ( a b
)n=an
bn
 EX: ( 2 5
)2=2
5
×2
5
=4
25
 ( 2 5
)2=22
52
=4
25

When an exponent is 1, the base remains the exact same.

a1 = a

When an exponent is 0, the outcome of the exponentiation of any kind of base will certainly always be 1, although somecontroversy surrounds 00 being 1 or unidentified. For many applications, defining 00 as 1 is convenient.

a0 = 1

Shown below is an instance of an argument for a0=1 making use of one of the abovementioned exponent legislations.

If an × am = a(n+m)Thenan × a0 = a(n+0) = an

Therefore, the only means for an to reprimary unadjusted by multiplication, and this exponent law to remajor true, is for a0 to be 1.

When an exponent is a portion wright here the numerator is 1, the nth root of the base is taken. Shown listed below is an instance via a fractional exponent wright here the numerator is not 1. It supplies both the dominion presented, as well as the dominance for multiplying exponents via like bases debated over. Keep in mind that the steustatiushistory.org can calculate fractional exponents, however they should be entered into the steustatiushistory.org in decimal create.

See more: Why Did The Student Eat His Math Exam Riddles, Q: Why Did The Student Eat His Homework It is also feasible to compute exponents with negative bases. They follow a lot the exact same rules as exponents via positive bases. Exponents via negative bases increased to positive integers are equal to their positive countercomponents in magnitude, however vary based on sign. If the exponent is an even, positive integer, the worths will be equal regardless of a positive or negative base. If the exponent is an odd, positive integer, the outcome will aget have the very same magnitude, however will be negative. While the rules for fractional exponents through negative bases are the same, they involve the use of imaginary numbers considering that it is not feasible to take any kind of root of an unfavorable number. An example is provided listed below for reference, but please note that the steustatiushistory.org gave cannot compute imaginary numbers, and any type of inputs that bring about an imaginary number will rerotate the result "NAN," signifying "not a number." The numerical solution is fundamentally the very same as the case through a positive base, other than that the number should be delisted as imaginary.